{"paper":{"title":"Subdifferential representation of convex functions on $X^*$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Duanxu Dai","submitted_at":"2017-11-28T07:51:58Z","abstract_excerpt":"In this paper, we obtain subdifferential representation of a proper $w^*$-lower semicontinous convex function on $X^*$ as follows: Let $g$ be a proper convex $w^*$-lower semicontinuous function on $X^*$. Assume that int dom $g$ $\\neq\\emptyset$ (resp. int (dom ($g^*|_X)$)$\\neq\\emptyset$). Then given any point $x_0^*$ $\\in$ D ($\\partial g\\cap X$) and $x^*$ $\\in$ dom $g$ (resp. $x^*\\in X^*$), we have $$g(x^*)=g(x_0^*)+\\sup\\{\\sum_{i=0}^{n-1}\\langle x_i,x_{i+1}^*-x_i^*\\rangle +\\langle x_n,x^*-x_n^*\\rangle \\},$$ where the above supremum is taken over all integers $n$, all $x_i^*\\in X^*$ and all $x_i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.10161","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}